A chord is drown through the focus of the parabola `y^(2)=6x` such than its distance from the vertex of this parabola is `(sqrt5)/(2)`, then its slope can be

Options

A. `(sqrt5)/(2)`
B. `(sqrt3)/(2)`
C. `(2)/(sqrt5)`
D. `(2)/(sqrt3)`

Answer

To solve the problem, we need to find the slope of a chord that passes through the focus of the parabola \( y^2 = 6x \) and is at a distance of \( \frac{\sqrt{5}}{2} \) from the vertex of the parabola.

### Step 1: Identify the parameters of the parabola
The given equation of the parabola is \( y^2 = 6x \). This can be compared with the standard form \( y^2 = 4ax \).

From the equation \( 4a = 6 \), we can find:
\[ a = \frac{6}{4} = \frac{3}{2} \]

### Step 2: Determine the focus of the parabola
The focus of the parabola \( y^2 = 6x \) is located at the point \( (a, 0) \):
\[ \text{Focus} = \left( \frac{3}{2}, 0 \right) \]

### Step 3: Set up the equation of the chord
Let the slope of the chord be \( m \). The equation of the chord passing through the focus \( \left( \frac{3}{2}, 0 \right) \) can be expressed as:
\[ y - 0 = m \left( x - \frac{3}{2} \right) \]
This simplifies to:
\[ y = mx - \frac{3m}{2} \]

### Step 4: Calculate the distance from the vertex
The vertex of the parabola is at the origin \( (0, 0) \). The distance \( D \) from the vertex to the line \( Ax + By + C = 0 \) can be calculated using the formula:
\[ D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For our line \( mx - y - \frac{3m}{2} = 0 \), we have:
- \( A = m \)
- \( B = -1 \)
- \( C = -\frac{3m}{2} \)

Substituting \( (x_1, y_1) = (0, 0) \):
\[ D = \frac{|m(0) - 1(0) - \frac{3m}{2}|}{\sqrt{m^2 + (-1)^2}} = \frac{\left| -\frac{3m}{2} \right|}{\sqrt{m^2 + 1}} = \frac{\frac{3|m|}{2}}{\sqrt{m^2 + 1}} \]

### Step 5: Set the distance equal to \( \frac{\sqrt{5}}{2} \)
We set the distance equal to \( \frac{\sqrt{5}}{2} \):
\[ \frac{3|m|}{2\sqrt{m^2 + 1}} = \frac{\sqrt{5}}{2} \]
Multiplying both sides by \( 2\sqrt{m^2 + 1} \):
\[ 3|m| = \sqrt{5} \sqrt{m^2 + 1} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 9m^2 = 5(m^2 + 1) \]
Expanding the right side:
\[ 9m^2 = 5m^2 + 5 \]
Rearranging gives:
\[ 9m^2 - 5m^2 - 5 = 0 \]
\[ 4m^2 - 5 = 0 \]

### Step 7: Solve for \( m^2 \)
\[ 4m^2 = 5 \implies m^2 = \frac{5}{4} \]
Taking the square root gives:
\[ m = \pm \frac{\sqrt{5}}{2} \]

### Conclusion
The possible slopes of the chord are:
\[ m = \frac{\sqrt{5}}{2} \quad \text{or} \quad m = -\frac{\sqrt{5}}{2} \]